well u have 3 inputs a b and c and a circuit that consists of only 2 not gates and any number of AND and OR gates u must now construct those gates inorder to get 3 out puts separetly which are a' b' and c'
that is the question and a hint would be is that it is necessary to use more than 20 gates.

You're not answering the question of what the circuit does. I could easily designate a',b',c' outputs the same as the inputs and use the two inverters in series to cancel each other out. Would you mind posting the problem as it is?

EDIT: Someone pointed out to me that a',b',c' refer to the inverted outputs of a,b,c. Now I understand the question.

I was bored so I made a rough outline for you to get started. I've uploaded and linked the picture, but since I renamed all your symbols in greek you will probably have to start from the beginning to use the picture i provided.

This problem is an excellent example of pretty difficult "easy" problems. Your professor (and other smarty pantses who have solved this) will promote it as a "novel" example, and it is, but this ignores the fact that the solution is quite involved. Indeed, anyone can solve this problem using very simple rules, but coming to the solution on your own requires some concerted effort.

The easiest place to start is to consider what the solution will look like. Take your initial variables, XYZ, and you need to end up with their complements: X', Y', Z'. Using DeMorgan's law, how can you obtain X (or it's complement) from a sum (OR) of other expressions? Write an expression for a single variable first:
X = (X+Y+Z)'(X+Y+Z')'(X+Y'+Z)'(X+Y'+Z')'
X' = X'Y'Z' + X'Y'Z + X'YZ' + X'YZ.

From this, you should be able to see that you need to construct (using your 2 given NOT gates) 2 types of expressions: one containing terms with 2 NOT'ed variables, and one containing terms with 1 NOT'ed variable. Try to build these, or at least work backwards from the awful drawing I made, and you should be able to see how the solution works. You will see that these will necessarily involve terms including all three, starting from basically X'Y'Z' and (X' + Y' + Z').